Question: Simplify; express your answer in exponential form. Assume $a\neq 0, p\neq 0$. $\dfrac{{(a^{4}p^{-4})^{5}}}{{(a^{-3}p^{-3})^{-5}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{4}p^{-4})^{5} = (a^{4})^{5}(p^{-4})^{5}}$ On the left, we have ${a^{4}}$ to the exponent ${5}$ . Now ${4 \times 5 = 20}$ , so ${(a^{4})^{5} = a^{20}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{4}p^{-4})^{5}}}{{(a^{-3}p^{-3})^{-5}}} = \dfrac{{a^{20}p^{-20}}}{{a^{15}p^{15}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{20}p^{-20}}}{{a^{15}p^{15}}} = \dfrac{{a^{20}}}{{a^{15}}} \cdot \dfrac{{p^{-20}}}{{p^{15}}} = a^{{20} - {15}} \cdot p^{{-20} - {15}} = a^{5}p^{-35}$